New View on Strong Nuclear Force 2

RealQM model of Helium nucleus consisting of 4 protons held together by electron cloud of  2 electrons.

This is a continuation of the previous post. We consider a RealQM model of the nucleus of an atom consisting of Z protons (each with charge +1) and N neutrons with in the normal case N=Z. Since each neutron can be viewed as a proton + an electron (with charge -1) the RealQM nucleus model consists of 2Z protons and Z electrons. 

We compare with a model of an atom ion with kernel charge +2Z surrounded by Z electrons, thus of charge +Z. With the electrons (for simplicity) homogenised into a single charge density $\phi (x)$ around the kernel (then without internal repulsion), the corresponding Hamiltonian takes the following form (in atomic units of length) 

• $-\frac{1}{2}\Delta_x - \frac{2Z}{\vert x\vert}$     (1)

where $\Delta_x$ is the Laplacian differential operator acting on a 3d space variable $x$. Assuming spherical symmetry it follows from the corresponding 1d Schrödinger equation that the bulk of the charge density has a width scaling with $\frac{1}{Z}$. In other words, the charge $Z$ sets the physical scale of the electron charge in terms of atomic units of length from a balance of kinetic energy and potential energy. The total energy then scales with $Z^2$.

Let us now try a model for the nucleus with Hamiltonian 

• $H=-\frac{1}{2}\Delta_x - K(x)$                           (2)

where $K(x)$ is the sum of potentials of the form $ \frac{1}{\vert x -x_i\vert}$, where the $x_i$ for $i=1,..., 2Z$ represent the positions of the 2Z protons of the kernel, with again $H$ acting on a single electron charge density $\phi (x)$ (without internal repulsion). We can alternatively view (2) as model of a molecule ion consisting of 2Z protons held together by an electron cloud of charge Z acting like glue connecting the protons. We compare with (1) with an electron cloud of charge Z surrounding a single nucleon of charge 2Z. 

As a model of a nucleus (2) represents a physical scale which is a factor of at least 1000 smaller than when viewed as a model of a molecule ion, in which the former case the electron appears as being compressed "inside" the nucleus instead of being "outside" as in (1). 

We can now play with the model (2) and vary Z as an "effective charge" setting a scale where the size of the electron matches the distance between the protons so that it can act as glue keeping protons together even under proton-proton repulsion. This scale can be found by minimizing the total energy normalised by $Z^2$.  Here you can play case of Helium nucleus (2P+2N) and here with a (hypothetical) Beryllium nucleus (4P+4N). 

This analysis suggests that it may be possible to replace the strong force supposed to keep an atomic nucleus together, by the electromagnetic force between protons and electrons acting on a smaller nuclear scale. Quite a bit of simplification if true...

To sum up, we are led to explore the possibility that protons and electrons can interact on two different scales: 

1. Electrons on atomic/molecule scale surrounding smaller scale nuclei. 
2. Electrons on smaller nuclear scale "compressed" with protons as neutrons and then acting like a glue between protons just like in a molecular ion. (It is not clear if "compressed" electrons inside a nucleus are subject to repulsion).

The scale difference is around a factor 1000 with thus the energy dissociation of one electron increasing from eV for atoms to MeV for nuclei. 

Summary so far as concerns the composition of a nucleus (connecting to Platons ideal solids):

• Hydrogen: 1P
• Deuterium: 1P+1N, shape: 1d linear
• Tritium: 1P+2N, 2d triangle (compare with H3+ tritium cation stable)
• Helium: 2P+2N, 3d tetrahedron
• Lithium: 3P+3N, octahedron
• Beryllium: 4P+4N, cube
• ...
• Coal: 6P+6N, icosahedron 
• ...
• Neon 10P+10N, dodekahedron
• ...